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I am struggling with a problem, where in the middle of my calculations I need to determine a function ##f(\alpha, \beta; x)##, namely the function of ##x## parametrized by ##\alpha## and ##\beta##, from the following equation

$$f\left(\alpha, \beta;\frac{1}{x}\right) = x^4f(\gamma, \delta; x).$$

So the ##f(1/x)## on the LHS and ##f(x)## on the RHS may differ in parametrization. Somehow I found that this equation admits the following solution

$$f(\alpha, \beta; x) = \left(\alpha + \beta x^a\right)^b $$

where ##a\cdot b = -4##. Although, this is probably not the most general solution.

So my question is what type of equation this one is? And how one should approach such equation?

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# What type of equation is this one?

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